Very interesting design concept there, one of the most interesing appliance designs I've seen in recent months if not longer.
Here's my wild guess as to how that one works.
You load the damp laundry in via the porthole in the front. The bottom of the machine is another hemisphere like the top dome, but is made of steel and is perforated at the bottom and around the upper circumference, just below the glass dome.
At the very center of the bottom is the opening where the warm air comes in at high velocity like the output of the blower port on a vacuum cleaner. At the periphery of the bottom hemisphere, just below where the top hemisphere begins, are the holes where the air exits. The air coming up from the bottom causes the load to fluff up into the upper dome, where it falls outward as it comes back down. This produces a tumbling action. As each article of clothing comes back down, it falls past the point where the air exits (thereby allowing lint to be sucked out), and slides down the sides of the lower hemisphere back to the bottom where the incoming warm air catches it and blows it back up to the top again, and so on.
While this is going on, the evaporated moisture at first causes the upper dome to fog up visibly. But as the load becomes more dry, the upper dome de-fogs, which the user can see as an indication that the clothes are drying. By the time the load is dry, it's moving more quickly and is taking up more room in the upper dome as it fluffs up, almost like popcorn. At the end of the cycle, you open the porthole in front to remove the load.
A few reasons why this wouldn't have caught on in the market. One, depending on the velocity of the air circulation, lighter articles could get stuck to the air exit holes by way of the exiting air pressure (think of something stuck at the end of a vacuum cleaner nozzle). Two, these things would require large boxes to store and ship, so they would take up more space in a warehouse and might also be susceptible to breakage of the dome, both factors adding cost. And less important but still relevant, three, unlike dryers with a flat top surface, you couldn't use the top as a counter or work surface to fold clothes or store a load from the washer before putting it into the dryer.
But even so, it would be an interesting idea to try to resurrect, especially with modern materials for the dome. And of course it could be done in an electric version as well as gas. It would be easy enough to make a scale model of this to test the design principles (particularly the air flow); someone here should try it. The absence of a rotating drum would make it quite simple: only a heater and a high-velocity blower. I'm willing to bet that the air wouldn't even need to be heated super-hot for this to work well; warm might be sufficient. (I'm tempted to build a test model myself, it's so simple; but for the lack of free time...)
Does this sound reasonable, or does anyone else have a guess as to how this works? Or is there patent information available?